SAT-SOLVING-EXAMPREP/Decision Heuristics.md

• Most important heuristic, as it can steer the entire solver

Types

• Static heuristics (pre-computed once)
  • Ex.: Natural Order
• Dynamic heuristics
  • Ex.: Dynamic Largest Individual Sum (DLIS) (chose most common variable + phase)
  • Expensive, needs (full) scanning of (all) clauses
• Second order dynamic heuristics
  • Ex.: VSIDS, ESIDS
  • Less expensive, good measurement

Look-ahead

• Instead of scanning static clause counts (DLIS) or scores (VSIDS), actually try each candidate variable
• For a candidate x: tentatively assign x and separately \overline x, run full [propagation (BCP)](Solvers#Boolean Constraint Propagation (BCP)) for each, and measure the effect (# clauses reduced/simplified, # new units)
  • If one polarity's propagation conflicts -> failed literal -> the other polarity is forced (same primitive as [Variable instantiation](Preprocessing_Inprocessing#Variable instantiation)/[Vivification](Preprocessing_Inprocessing#Vivification / Distillation))
• Pick the variable (and often the polarity) that maximizes the combined look-ahead effect of both branches (e.g. product/sum of the two reduction counts, as in march/kcnfs)
• Very informative but expensive (full propagation per candidate, both polarities) -> basis of dedicated look-ahead solvers, rather than per-decision use inside CDCL

Cut heuristic

• Find a variables that 'cuts' the CNF into parts without overlapping variables
• Solve parts separately (for all assignments of the cut)
• Divide & Conquer approach

Horn Form

Definition

• A clause is positive if it contains at least 1 positive literal, otherwise it's negative
• A Horn clause is a clause with at most one positive literal
• A CNF is in Horn Form if all clauses are Horn clauses

Facts

• If all clauses in a Horn Form CNF are positive there is a minimal satisfying assignment
  • This assignment can be found using BCP
• The minimal satisfying assignment of the positive part of a Horn Form CNF IFF it is SAT
  • I.e. if not it is UNSAT

Usefulness/ Heuristic

• DP can transform a non Horn from CNF into one by choosing variables from clauses with more than one positive literal (non horn clauses) and doing resolution on them.
• [BCP](SAT_in_general#Special Case Unit Resolution) runs through a HORN CNF in polynomial time

Variable State Independent Decaying Sum (VSIDS)

• Count occurrence of variables in conflict clauses
• Priority Queue is updated, and VSIDS is taken from it
• Decay/ re-score values after i conflicts (Chaff: f = 1/2, i = 256)
  • multiply all by some f < 1
  • Emphasizes literals contributing to recent conflicts

Additional idea:

• Use VSIDS only as secondary heuristic, primarily try to satisfy must recently added clauses

Normalized VSIDS (NVSIDS)

• Keep the score between [0, 1]
• Pick f = 0.95 and calculate score like this: new_score = old_score * f + in_conflict? (1 - f) : 0
• You can defer the re-scoring to later, i.e. only update variables in conflict and then re-score all later on to save time
• Can also Bump resolved literals on conflict level, not just learned lits (good for heuristic?!)

Exponential VSIDS (EVSIDS)

• Bump by exponential value g^i = \frac{1}{f}^i
  • new_score = old_score + (1/f)^i
• EVSIDS is just NVSIDS but without scaling

\frac{f^{-n}}{1-f} \cdot \text{NVSIDS}

Decision heuristic Summary table

• Old score: s
• decay 0 < f < 1

STRATEGY	Bumped	not-bumped
STATIC	s	s
INC	s + 1	s
SUM	s + i	s
VSIDS	s / 2 + 1	s / 2	Only half every so often (every 256)
NVSIDS	f \cdot s + (1-f)	f \cdot s
EVSIDS	s + f^{-i}	s
AVG	(s+i)/2	s	average conflict-index decision scheme
VMTF	i	s	Variable Move to Front

• Turns out AVG (moving average) and EVSIDS are very good

Variable Move to Front

• Cheap to implement, aggressive